Tic tac toe cubed

Question

The cube shaped game board is made from 27 transparent unit cubes. Similar to regular version of tic tac toe you are trying to arrange your marbles into lines of 3. How many different winning lines are there in a 3x3x3 cube?

Answer 1

The 3-D cube is an ideal example for computer data structures and algorithms.

A 3 x 3 x 3 cube has 6 sides, each with 9 little cubes. Each of the 6 sides on a little cube may connect to none or one other cube. All solution chains begin and end with a surface cube. Solution chains connect cubes in a straight line, moving either from side to side or else from corner to corner (that is, in a straight line).

If the 3-D coordinates of each of the 27 little cubes are (x,y,z), then

   (1,1,1) links to (2,1,1) links to (3,1,1)     [along x-axis]

   (1,1,1) links to (1,2,1) links to (1,3,1)     [along y-axia]

   (1,1,1) links to (1,1,2) links to [1,1,3]     [along z-axia]

   (1,1,1) links to (2,2,2) links to (3,3,3)     [diagonal to opposite corner of cube]

…

 

Now, all solution chains begin or end with one of the surface cubes [that is, (2,2,2) never starts or ends a solution chain]. Also, reversed solution chains should be counted only once.

So, can we list the number of chains that start on a given cube?   Yes !!

    Corner - there are 8 corners - 7 chains (surface and inside diagonal) -- 56 chains

    Center of edge -- there are 12 of these - each starts 3 chains  -   36 chains

    Center of surface -- there are 6 of these - each starts 1 chain  -  6 chains

    Center of big cube -- no chains

                       Total -- 98

This list follows a pattern that a computer may find using exhaustive enumeration (consider ALL possibilities).  To count solutions and their reverse only once, 98/2 = 49.

There are 49 straight lines that solve the 3D tic-tac-toe.

Answer 2

The way I read this, we are to find all of the unique lines of 3. We can use an XYZ coordinate system to describe the lines.

Looking at the X=1 face (plane), describe the 8 possible lines as follows:

(Note that "n" means "any" value)

1 1 n (the line Y=1 on the plane X=1

1 2 n

1 3 n

1 n 1 (the line Z=1 on the plane X=1)

1 n 2

1 n 3

1 Z n (the line Y=Z on the plane X=1)

1 n Y

Repeat this for the X=2 and X=3 planes, and we have 24 unique lines.

Next, look at the planes defined by a Y value:

Only the ones in bold are new ( 15 new)

1 1 n

2 1 n

3 1 n

n 1 1

n 1 2

n 1 3

Z 1 n

n 1 X

repeat for Y = 2 and Y = 3

And for the Z planes (new in bold ---6 are new):

1 n 1

2 n 1

3 n 1

n 1 1

n 2 1

n 3 1

Y n 1

n X 1

(Repeat for Z = 2 and Z = 3)

So far, we have 24 + 15 + 6 = 45 unique lines

Are there more???

EDIT:

I omitted the planes defined by the diagonal of the faces, However, even though these planes are unique, all of the lines on them are redundant, EXCEPT the lines between the most distant corners of the cube. There are 4 of these, putting the total at 49